Reliability Evaluation and Diagnostics with Propositional Directed Acyclic Graphs

The theories of reliability and diagnostics, despite their close relationship, have been mostly treated as two separate fields in the literature. However, it has been shown in the past few years that their dual character can be exploited in the context of modular systems. Such systems have a great practical impact on reliability computation and diagnostics, since their structure can be used to significantly reduce the computational effort needed for their evaluation. It is also known that the underlying structure function can be represented in a compact way by Propositional Directed Acyclic Graphs (PDAGs), which form a powerful Boolean representation language. Recently, PDAGs have been also applied successfully in the context of network reliability. The present paper continues this line of research and proposes several extensions. First, we introduce a unifying formal model for system description, reliability evaluation, and diagnostics. Besides the formal specification of modular systems and reliability networks, the model allows networks and modules to be combined arbitrarily, leading to so-called hybrid systems. The underlying computational machinery provided by PDAGs allows to evaluate reliability in a convenient and efficient way. Second, we provide concise algorithms for both generating structure functions with respect to network reliability problems as well as for PDAG manipulations, and we also examine their complexity. Third, we show that posterior probabilities of system elements can be efficiently computed using PDAGs for diagnostic purposes, in particular in modular systems.